Ellipse more properties
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This document discusses properties of ellipses. It defines an ellipse as the set of all points where the sum of the distances from two fixed points (foci) is a constant. Examples are provided to demonstrate how to find the standard equation of an ellipse given its foci or other properties, as well as how to determine points like the center, vertices, and covertices. Applications involving tunnel and bridge dimensions are also presented.
Ellipse more properties
- 2. REMEMBER THIS! An ellipse is the set of all points on the plane, the sum of whose distances from two fixed points is a constant.
- 3. MORE PROPERTIES OF AN ELLIPSE Some ellipses have their foci aligned vertically, and some have centers not at the origin. Their standard equations and properties are given in the next slide. where a > b and 𝑐 = 𝑎2 − 𝑏2 .
- 6. Example 1. Give the coordinates of the center, foci, vertices, and covertices of the ellipse with the given equation. Sketch the graph, and include these points. (a) (𝑥+3)2 24 + (𝑦−5)2 49 = 1 (b) 9x2 + 16y2 - 126x + 64y = 71 Example 2. Find the (standard) equation of the ellipse whose foci are F1(3,0) and F2(3,0), such that for any point on it, the sum of its distances from the foci is 10.
- 7. Solution. (1.a) From a2 = 49 and b2 = 24, we have a = 7, b =2 6 ≈ 4.9, and c = 𝑎2 − 𝑏2 = 5. The ellipse is vertical. center: (-3,5) foci: F1(-3,0), F2(-3,10) vertices: V1(-3,-2), V2(-3,12) covertices: W1(−3 − 2 6,5) ≈(7.9,5) W2(−3 + 2 6,5) ≈(1.9,5)
- 8. Example 3. The foci of an ellipse are (-3,-6) and (-3,2). For any point on the ellipse, the sum of its distances from the foci is 14. Find the standard equation of the ellipse. Example 4. A tunnel has the shape of a semi ellipse that is 15 ft high at the center, and 36 ft across at the base. At most how high should a passing truck be, if it is 12 ft wide, for it to be able to fit through the tunnel? Round off your answer to two decimal places.
- 9. Try This! 1. Give the coordinates of the center, foci, vertices, and covertices of the ellipse with the given equation. Sketch the graph, and include these points. 36x2 + 20y2 -144x + 120y - 396 = 0 2. The arch of a bridge is in the shape of a semiellipse, with its major axis at the water level. Suppose the arch is 20 ft high in the middle, and 120 ft across its major axis. How high above the water level is the arch, at a point 20 ft from the center (horizontally)?